Square Roots

Square roots...oh so useful...this may take awhile..

So, what is a square root?  By algebraic definition,

  _{_}
√a = b if b² = a

Now, a number can obviously have 2 square roots - a positive and negative one.  This is because positive * positive = positive, and negative * negative = positive.  In the same way, negative numbers have no real number square roots.

(By the way, if the page looks a little choppy, it's because it is hard to get the square root symbol to look right on computers).

--Some numbers are perfect squares.  This means that they have integer square roots.  Some of these numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.

                                      _                                                _{_}
--Also, the expression √a is called a RADICAL.  The √    is called the RADICAL SIGN.

--Sometimes radicals may not be square roots.  In a cube root, the root is a 3.  If the root is not a square root, the number of the root will be displayed where the "n" is here:

_{n _}
 √a

--The square root's number is a 2, but it is assumed to be that if there's nothing there.  As a rule, negative numbers do have real number roots, as long as the root isn't even.  For example, the cube (3) root of -27 is -3, because -3*-3*-3 = -27.

--Now, most numbers are not perfect squares, and you often have to work with these numbers' square roots, especially in right triangles (next section of the tutorial).  There are a few ways to approximate square roots...sometimes you even have a square root table to work with.  However, say you want to find (without a calculator):
  __
√51

To find this, use the divide-and-average method.  We will approximate this to the nearest tenth.
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First, think of which 2 numbers this lies between.  The square of 7 is 49, and the square of 8 is 64, so √51 lies between 7 and 8.

Next, take an educated guess to the place value that you are finding for of what it might be...51 is much closer to 49 than 64, so let's make an estimate of 7.2

Now we divide...
51/7.2 ≈ 7.08...

Now, the tenth places of the divisor and quotient are not the same, so we find the hundredth place of the quotient (which we did).

Now comes the averaging part.  We find the average of the divisor and quotient, because they, multiplied together, will give us 51, and we want to get them as close as possible, if not the same.

7.08 + 7.2 = 14.28
14.28/2 = 7.14, which we try again to divide.
51/7.14 ≈ 7.142...                                                 __
The tenth places are the same, so we can say √51 ≈ 7.1 to the nearest tenth.

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